Signal Reconstruction from Projections

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We now know how to project a signal onto other signals. We now
need to learn how to reconstruct a signal from its projections onto different
vectors , . This will give us theinverse DFT operation
(or the inverse of whatever transform we are working with).

As a simple example, consider the projection of a signal
onto the rectilinear coordinate axes of .
The coordinates of the projection onto the th
coordinate axis are simply . The projection along coordinate axis has
coordinates , and so on. The original signal is then
clearly the vector sum of its projections onto the
coordinate axes:

To make sure the previous paragraph is understood, let’s look at
the details for the case . We want to project an arbitrary two-sample signal
onto the coordinate axes in 2D. A coordinate axis can be
represented by any nonzero vector along its length. The horizontal
axis can be represented by any vector of the form
while the vertical axis can be represented by any vector of the
form . For maximum simplicity, let’s choose the positive unit-length
representatives:

The projection of onto is by definition

Similarly, the projection of onto is

The reconstruction of from its
projections onto the coordinate axes is then the vector sum of
the projections:

The projection of a vector onto its coordinate axes is in some
sense trivial because the very meaning of the coordinates is
that they are scalars to be applied to the coordinate vectors in order to form an arbitrary vector as a linear combinationof the coordinate
vectors:

Note that the coordinate vectors are orthogonal.
Since they are also unit length, , we say
that the coordinate vectors are orthonormal.

What’s more interesting is when we project a signal onto a set of vectors other than the coordinate set.
This can be viewed as a change of coordinates in .
In the case of the DFT, the new vectors will be chosen to be
sampled complex sinusoids.